Oriented Matroids: The Power of Unification
3. Arrangements of lines
The diagram above shows a collection of 5 lines, which you should think of as going off to infinity rather than being line segments. Furthermore, the arrangement above is a simple arrangement which divides the plane into bounded regions with 3 sides, 4 sides, and 5 sides. In fact, there are four 3-gons, one 4-gon, and one 5-gon. How should we treat the unbounded regions? One approach is to think of the two regions labeled R1 and R2 as really being the same region. Think of the two lines which form R2 and move off the top of the diagram as coming back up from the bottom and, thus, defining a region which is the same as R1. Let us refer to this one region as R. How many sides should we say R has? Since the pieces of R are bounded by 4 lines, it is natural to call R a 4-gon. In this perspective we can think of being involved with a geometry whichlacks parallelism for lines and is known as the (real) projective plane. We have points, lines and the following rules (among others):
So far we have thought of S as arising from intersections of lines in an arrangement of lines. However, suppose we consider any set S* of points (not all on a line) in the plane, which may or may not arise as the set of points where lines in an arrangement meet. Now consider the number of points of S* that lie on each line that is determined by a pair of points in S*. Will there always be such a line with exactly two points on it? This problem, in essence due to J. J. Sylvester and revitalized by Paul Erdös and Tibor Gallai, has been widely explored and has many ramifications.
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